CRYSTALLOGRAPHIC ASPECTS OF GEOMETRICALLY- NECESSARY …li.mit.edu/Stuff/M2/Upload/Arsenlis/arsenlis1999.pdf · CRYSTALLOGRAPHIC ASPECTS OF GEOMETRICALLY-NECESSARY AND STATISTICALLY-STORED

CRYSTALLOGRAPHIC ASPECTS OF GEOMETRICALLY-

NECESSARY AND STATISTICALLY-STORED

DISLOCATION DENSITY

A. ARSENLIS and D. M. PARKS{Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA

01239, U.S.A.

(Received 14 October 1998; accepted 22 December 1998)

AbstractÐClassical plasticity has reached its limit in describing crystalline material behavior at the micronlevel and below. Its inability to predict size-dependent e�ects at this length scale has motivated the use ofhigher-order gradients to model material behavior at the micron level. The physical motivation behind theuse of strain gradients has been based on the framework of geometrically-necessary dislocations (GNDs).A new but equivalent de®nition for Nye's dislocation tensor, a measure of GND density, is proposed,based on the integrated properties of dislocation lines within a volume. A discrete form of the de®nition isapplied to redundant crystal systems, and methods for characterizing the dislocation tensor with realizablecrystallographic dislocations are presented. From these methods and the new de®nition of the dislocationtensor, two types of three-dimensional dislocation structures are found: open periodic networks which havelong-range geometric consequences, and closed three-dimensional dislocation structures which self-termi-nate, having no geometric consequence. The implications of these structures on the presence of GNDs inpolycrystalline materials lead to the introduction of a Nye factor relating geometrically-necessary dislo-cation density to plastic strain gradients. # 1999 Published by Elsevier Science Ltd on behalf of Acta Metal-lurgica Inc. All rights reserved.

Keywords: Metals, Dislocations, theory, Mechanical properties, plastic

1. INTRODUCTION

Recent experiments have shown a size dependence

in the mechanical behavior of materials at the sub-

micron to micron level. Torsion of thin wires ran-

ging in diameter from 12 to 170 mm has shown that

there is increased torsional hardening as the diam-

eter of the specimens decreases [1]. Bending of thin

beams ranging in thickness from 12.5 to 50 mm has

shown that there is increased material hardening in

bending as the thickness of the beams decreases [2].

Micro-indentation experiments conducted at these

length scales have shown similar phenomena. The

measured indentation hardness of a crystal is found

to increase as the depth of the indentation decreases

from 10 to 1 mm [3, 4]. All of these experiments

have shown that the apparent material hardening

increases as the size of the specimen decreases.

Classical plasticity theory has been unable to

account for the observed phenomena because it has

no internal length scale in its formulation. All of

the experiments which highlight these material size

e�ects have associated with them large strain gradi-

ents with respect to the overall deformation. As a

result, higher-order gradient theories have been pro-

posed which incorporate a characteristic material

length scale [5, 6] as a material property. The in-ternal length scale has been introduced through

dimensional arguments, based on the formulationof particular theories, and has been empirically

inferred to be on the order of sub-microns tomicrons [1, 2, 6].

The strain gradient theories have been physicallymotivated by developments in dislocation mech-

anics. In particular, the framework of geometri-cally-necessary dislocations (GNDs) ®rst introduced

by Nye [7] and furthered by Ashby [8] andKroÈ ner [9] has given a physical basis for strain-gra-dient-dependent material behavior. In descriptions

of dislocations in crystals, dislocations could be sep-arated into two di�erent categories, geometrically-

necessary dislocations which appeared in strain gra-dient ®elds due to geometrical constraints of the

crystal lattice, and statistically-stored dislocations(SSDs) which evolved from random trapping pro-

cesses during plastic deformation [8]. AlthoughGNDs have provided the motivation for these the-

ories, GNDs have been loosely interpreted, andcontinuum theories have mostly been concernedwith ®nding relationships between the invariants of

strain gradients and the mechanical behavior ofmaterials [10, 11].

The length scale dependence suggests that theunderlying properties of the crystal lattice should

Acta mater. Vol. 47, No. 5, pp. 1597±1611, 1999# 1999 Published by Elsevier Science Ltd

On behalf of Acta Metallurgica Inc. All rights reservedPrinted in Great Britain

1359-6454/99 $20.00+0.00PII: S1359-6454(99)00020-8

{To whom all correspondence should be addressed.

1597

play a larger role in the formulation of continuum

strain gradient theories, and a more rigorous in-terpretation of GNDs, one which accounts for crys-

talline anisotropy, be implemented. As a ®rst step

in describing GND populations in a polycrystallinematerial, the properties of GNDs and SSDs in

single crystals are investigated. This article discusses

the physical arguments for the appearance of dislo-cations due to strain gradients and introduces a

new de®nition for Nye's tensor, a quantity used to

measure GND density, based on the integratedproperties of dislocation lines within a volume.

With this new de®nition, global properties of the

two dislocation categories are immediately appar-ent. A discretized version of the integral relation is

developed to investigate the geometric properties of

dislocations in crystals which can accommodategeometrical constraints with di�erent dislocation

arrangements. Two methods are introduced to cal-

culate the GND density in these redundant crystals,and new SSD arrangements are discovered. The

methods are applied to the face-centered cubic

(f.c.c.) lattice to demonstrate the properties of theGNDs in this discrete formulation. The results of

the single crystal investigation have important con-

sequences on the presence of GNDs in polycrystal-line materials, and a Nye factor is introduced which

relates macroscopic strain gradients to scalar

measures of GND density in polycrystalline ma-terials.

Fig. 1. Schematic process (a±e) through which geometrically-necessary edge dislocations accumulate.

ARSENLIS and PARKS: CRYSTALLOGRAPHIC ASPECTS OF DISLOCATION1598

2. CRYSTALLOGRAPHIC DISLOCATIONS FROMSTRAIN GRADIENTS

Gradients in the plastic strain within crystalline

materials give rise to dislocations in order to main-

tain continuity in the crystal. Furthermore, with

knowledge of the crystalline orientation in relation

to the strain gradient, the type of dislocation

needed to maintain lattice continuity is also speci-

®ed. The graphical arguments for the existence of

these dislocations presented in this section are

based on the two-dimensional constructs of

Ashby [8] but have been extended to three-dimen-

sions.

Consider the schematic in Fig. 1(a)±(e) of a

simple crystal undergoing single slip on slip system

``a''. The coordinate reference frame is set such that

sa is a unit vector in the slip direction, na the slip-

plane unit normal, and ma � sa � na. Imagine thatthe material can be separated into three sections,

and each section can be deformed independently of

the others. Through expanding dislocation loops,

the respective sections are plastically deformed such

that the plastic strain increases linearly in the slip

direction. When the dislocation loops reach the

boundaries of each section, the screw portions

reach free boundaries and exit the material, but the

edge portions of the loops encounter ®ctitious in-

ternal boundaries and remain as dipoles spread to

either side of each section. The sections are then

forced back together, and there are negative edge

dislocations which do not annihilate, but remain in

the material, leading to lattice curvature.

Mathematically, the relationship between the

plastic strain gradient on a slip system and the edge

dislocation density takes the following form:

raGN�e�b � ÿrga � sa � ÿga,ksak �1�

where raGN�e� is the geometrically-necessary positive

edge dislocation density and b the magnitude of the

Burgers vector on slip system a.

A similar construction can be created to illustrate

the presence of screw dislocations due to strain gra-

dients. The schematic in Fig. 2(a)±(e) shows the

same single slip system material as in Fig. 1(a)±(e).

In this ®gure, the material is again separated into

three sections, and each section is deformed inde-

pendently of the other. Through expanding dislo-

cation loops, the sections are plastically deformed

such that the plastic strain increases linearly in the

Fig. 2. Schematic process (a±e) through which geometrically-necessary screw dislocations accumulate.

ARSENLIS and PARKS: CRYSTALLOGRAPHIC ASPECTS OF DISLOCATION 1599

ma-direction. When the dislocation loops reach theboundaries of each section, the edge portions reach

free boundaries and exit the material, but the screwsegments encounter ®ctitious internal boundariesand remain as dipoles spread to either side of each

section. In the ®gure shown, the negative screw dis-locations are illustrated, but there are also an equalnumber of positive screw dislocations, not shown,

on the back (hidden) side of each section. When thesections are forced back together, there are positivescrew dislocations which do not annihilate, remain-

ing in the material and causing the lattice to warp.The relationship between the plastic strain gradi-

ent on a slip system and the screw dislocation den-sity takes the following form:

raGN�s�b � rga �ma � ga,kmak �2�

where raGN�s� is the geometrically-necessary positivescrew dislocation density and b the magnitude ofthe Burgers vector on slip system a.

These developments clearly show that gradientsof plastic strain lead to the accumulation of dislo-cations. Furthermore, depending on the direction ofthe gradients in relation to the crystalline geometry,

the type of dislocations needed to maintain latticecontinuity can be speci®ed, independent of themechanism which caused the strain gradient. The

plastic deformation in each example was accom-plished through expanding dislocation loops, butthe gradients of the plastic deformation were in

di�erent directions, leading to the accumulation ofeither geometrically-necessary edge or screw dislo-cations. The existence of the dislocations wasnecessary to maintain lattice continuity, and led to

distortions of the the crystal lattice. Generally, thedislocation state associated with lattice distortioncan be described in terms of a second-order tensor,

to be discussed in the next section.

3. NYE'S TENSOR AND CONTINUOUSLY-DISTRIBUTED DISLOCATION DENSITY

In 1953, Nye introduced a dislocation tensor

quantifying the state of dislocation of a lattice [7].Nye's tensor, aij, is a representation of dislocationswith Burgers vector i and line vector j. Consideringcontinuously-distributed dislocations, Nye's tensor

quanti®es a special set of dislocations whose geo-metric properties are not canceled by other dislo-cations in the crystal. Consider the volume element

shown in Fig. 3, which is a section of a crystal con-taining two edge dislocations threading through thevolume. The most rigorous manner to describe the

dislocation state in the volume would be to charac-terize the lines by two Dirac delta functions ofstrength b in space; however, such a quanti®cation

of dislocation density becomes overwhelming giventhe densities involved in plastic deformation pro-cesses (r�: 1012=cm2) [12]. Allowing such point den-sities to become continuously-distributed within a

volume creates a more compact way of describing

dislocations in space, but a result of this process isthat some individual dislocation information is lost.

If the reference volume element over which dislo-cation properties will be continuously-distributed is

taken to be the entire volume in Fig. 3, the net

Nye's tensor of the element is zero because the dis-location density, as drawn, consists of two dislo-

cations with common tangent line vector but

opposite Burgers vector: these form a dislocation

dipole. When the properties of each dislocation seg-ment are distributed uniformly over the entire

volume, the exact positions of the original dislo-

cations are no longer relevant. In terms of Nye'stensor, an equivalent form would be to place the

two dislocations on top of one another, allowing

them to annihilate, leaving behind no dislocationdensity in the element. In any continuously-distribu-

ted dislocation formulation, individual dislocation

dipoles, planar dislocation loops, and other three-

dimensional self-terminating dislocation structuresfully contained within the reference volume make

no net contribution to Nye's tensor. The three-

dimensional self-terminating dislocation structuresmentioned here will be further developed in later

sections. These redundant structures which make no

contribution to Nye's tensor are considered statisti-

cally-stored dislocations (SSDs), and are believed toresult from plastic deformation processes [8]. Once

individual dislocation segments are considered to be

uniformly distributed within a reference volume,Nye's tensor measures the non-redundant dislo-

cation density within the volume. These non-redun-

dant dislocations are believed to result from plasticstrain gradient ®elds, as demonstrated in the pre-

vious section, and have geometric consequences on

the crystal lattices. As a result, they are commonly

referred to as geometrically necessary dislocations(GNDs).

Fig. 3. An edge dislocation dipole in a volume elementused to de®ne Nye's tensor.


Nye's tensor can easily be calculated for a volume

element by a line integral over all dislocations

within the volume. If b is the Burgers vector of a

dislocation with local unit tangent line direction t,

Nye's tensor, aij, can be de®ned as

aij � 1

V

�L

bitj ds �3�

where V is the reference volume, ds an element of

arc length along the dislocation line, and L the

total length of dislocation line within V. Nye's ten-

sor becomes a summation of the integrated proper-

ties of all the individual dislocation line segments in

the volume. This integral relation also has the prop-

erty of averaging the dislocation properties within

the volume, thus converting clearly distinct dislo-

cation lines into a uniformly distributed property

within the volume. If each of the dislocation line

segments is considered as a separate entity with

constant Burgers vector, the de®nition of Nye's ten-

sor can be rewritten as

aij � 1

V

Xx

bxi

�l

txj dsx �4�

where l is the length of a dislocation segment of

type x. Inspection of this summation of integrals

immediately shows global properties of Nye's ten-

sor. Consider Fig. 4 of a dislocation threading

through a reference volume element used to de®ne

Nye's tensor. If equation (4) is used to evaluate

Nye's tensor for this element, the result becomes

aij � 1

Vbi�x�j ÿ xÿj � �5�

where xÿ and x+ are the positions of the starting

and stopping points of the dislocation line segment,

respectively. Equation (5) states that the only infor-

mation from each dislocation line segment needed

to calculate its contribution to Nye's tensor is its

Burgers vector and two endpoints. The path that

the dislocation line makes between these two points

has no e�ect on the evaluation of Nye's tensor.

Returning to Fig. 4, for the purpose of evaluating

Nye's tensor, the curved dislocation line could be

replaced by the dashed straight dislocation line hav-

ing the same geometric properties. Equation (5)

may be rewritten to re¯ect this substitution, with

the result that it is the average tangent vector, �t,

and the secant length, �lx, which are needed to calcu-

late aij:

aij � 1

V

Xx�l

xbxi �txj �6�

where

�txj ��x�j ÿ xÿj ��x�k ÿ xÿk ��x�k ÿ xÿk �

p �7�

and

�lx �

��x�k ÿ xÿk ��x�k ÿ xÿk �

q: �8�

Using the description of dislocation density as line

length in a volume, the summation of geometric dis-

location lengths, �lx, in a reference volume, V, can

be replaced by a summation of geometric dislo-

cation density, rxGN, in the volume

aij �Xx

rxGNbxi�t xj �9�

where

rxGN ��lx

V: �10�

Of course, the dislocation density described in

equation (9) is not the total dislocation density of

any arbitrary dislocation line segment, but it is the

portion of the total dislocation density which has

geometric consequences. The remaining density of

the total line, which has no geometric consequence,

must be considered statistical in character. For an

arbitrary dislocation line segment, x, with a total

density in a reference volume, V, de®ned by

rx � 1

V

�l

dsx �11�

the portion of the total density which has no geo-

metric consequence, and which would therefore be

statistical in nature, would be

rxSS � rx ÿ rxGN �12�where the subscript SS denotes that the dislocation

density is considered statistically-stored. With this

decomposition of total dislocation density, an arbi-

trary line threading through a reference volume el-

ement as in Fig. 4 may be separated into that

portion of the total density which has geometricFig. 4. A dislocation line threading through a reference

volume element used to de®ne Nye's tensor.


e�ects, rGN, and the portion of the total density

which does not, rSS.Another result of the de®nition of Nye's tensor

which becomes immediately apparent through

equation (5) is that closed dislocation loops of con-

stant Burgers vector have no net geometric conse-

quence; i.e.

aij � 1

Vbi

Il

tj ds � 0ij: �13�

As a generalization of this property, any dislocation

network structure which is topologically closed and

is entirely contained within the reference volume

also has no net contribution to Nye's tensor.

In this formulation of Nye's tensor, the size of

the reference volume element over which the density

is averaged plays a crucial role in de®ning the ten-

sor. Consider again Fig. 3 of the dislocation dipole.

If the reference volume elements were taken to be

smaller than the entire volume shown, such that

both dislocations did not populate the same el-

ement, Nye's tensor within each sub-volume would

change, becoming non-zero. In the limit as the

volume elements used to de®ne the dislocation state

become di�erential, Nye's tensor tends toward two

delta functions that exactly describe the dislocation

state. Selection of an appropriate reference volume

element must take into account the scale of the geo-

metric e�ects to be captured. A volume element

that is too large with respect to the geometric con-

straints may miss the existence of important geome-

trically-derived dislocations in one portion of the

element that, when averaged with other dislocations

in the same element, create no net Nye's tensor.

Conversely, a volume element which is too small

may become too computationally intensive (too nu-

merous) to manage, and may begin to reach length

scales where dislocation density can no longer be

considered continuously distributed, and discrete

dislocation mechanics must be incorporated.

The de®nition of Nye's tensor proposed here is

based on the description of dislocation density as

line length in a reference volume. In Nye's original

formulation of the dislocation tensor, dislocation

density was described as a number density of lines

piercing a plane. He de®ned the tensor in the fol-

lowing manner:

aij � nbitj �14�where n was the number density of dislocation lines

with Burgers vector, b, crossing a unit area normal

to their unit tangent line vector, t. This expression

closely resembles the de®nition of the tensor in

equation (9). In the procedure by which Nye

described the dislocation tensor, the dislocations

which constituted it were considered to be continu-

ously distributed, and the tangent line vectors were

implicitly constant. The discrete case, in which a

material is segmented into volume elements, was

not considered in the original formulation. The ex-pression proposed here is a generalization whichcan be applied to arbitrary dislocation arrange-

ments, and the two de®nitions are equivalentbecause the dislocations which contribute to thegeometric density in the bulk must pierce the sur-

face that encloses the reference volume, as a conse-quence of equation (13). Consider Fig. 5(a) whichrepresents a relatively simple dislocation arrange-

ment containing two dislocation junctions threadingthrough the volume. By applying equation (3) tothe system, it can be shown that a geometricallyequivalent dislocation arrangement would be a

single straight dislocation segment as shown inFig. 5(b). This results from the property thatBurgers vector is conserved through line segments

and their junctions. Applying both equations (3)and (4) to the simpli®ed structure in Fig. 5(b) yieldsthe same value for Nye's tensor. The de®nitions are

equivalent because of the stereological relationshipbetween the two descriptions of dislocation density.

4. LATTICE-GEOMETRIC CONSEQUENCES OFNYE'S TENSOR

Plastic strain gradients necessarily lead to theexistence of dislocations in crystalline materials to

Fig. 5. A simple dislocation network with two junctionsthreading through a reference volume element used to de-®ne Nye's tensor (a), and its corresponding geometric ®n-

gerprint (b).


maintain lattice continuity. Nye's tensor provides a

measure of these geometrically-derived dislocations,

and a non-zero Nye's tensor leads to lattice curva-

ture, neglecting elastic strain gradients in the ma-

terial. Introduction of dislocations of the same type

into a crystal causes the lattice to curve and gener-

ally warp, and a tensor quantifying such lattice cur-

vature must be closely related to Nye's tensor. The

curvature tensor, kij, is de®ned as a small right

handed lattice rotation of magnitude dW about the

i-axis for a unit change of position of magnitude dxin the j-direction:

dWi � kijdx j: �15�Nye's tensor relates the GND density to lattice cur-

vature in the following manner:

kij � ÿaji � 1

2djiakk �16�

which is the same result that Nye derived but with

a di�erent sign convention [7]. There is also a con-

tribution from elastic strain gradients to the total

lattice curvature. A complete and rigorous deri-

vation of the relationship between strain gradients,

curvature, and the new de®nition of Nye's tensor is

presented in this section.

The following derivation closely mirrors a pre-

vious analysis by Fleck and Hutchinson [10]. The

displacement gradient, ui,j, can be additively decom-

posed into the plastic slip tensor, gij, the skew lattice

rotation tensor, fij, and the symmetric elastic strain

tensor, eelij , as shown in equation (17):

ui,k � gik � fik � eelik: �17�The curl of the displacement gradient vanishes

because of the symmetry of the second gradient;

this also implies that deformation occurs in such a

way that the body remains simply connected. Such

a procedure, when performed on equation (17), pro-

duces

epjkui,kj � epjkgik,j � epjkfik,j � epjkeelik,j � 0pi �18�where epjk are the cartesian components of the alter-

nating tensor. The components of the lattice ro-

tation tensor, fik, can be written in terms of the

lattice rotation vector, Wl, according to

fik � eilkWl: �19�Substitution of equation (19) into equation (18)

gives

epjkgik,j � epjkeilkWl,j � epjkeelik,j � 0pi: �20�Using the de®nition of klj � Wl,j implied by

equation (15), inversion of equation (20) gives

kpi � epjkgik,j ÿ1

2dpiesjkgsk,j � epjkeelik,j: �21�

Since plastic deformation is a result of the slip on

crystallographic planes, gij can be written as a sum

of crystallographic shears such that

gik �Xa

gasai nak �22�

where sa is the slip direction and na the slip-plane

normal direction of slip system a on which a plastic

shear, ga, has occurred. Applying the curl operation

inside the summation yields

epjkgik,j �Xa

epjkga,jsai n

ak �

Xa

ga,jsai �saj ma

p ÿmaj s

ap� �23�

where the last step follows from the de®nitionma � sa � na. Using equations (1) and (2), the right

side of equation (23) can be replaced by dislocation

densities:

epjkgik,j �Xa

ÿ raGN�e�basai m

ap ÿ raGN�s�b

asai sap: �24�

Dislocation densities are considered to be de®ned as

length of line in a reference volume, which allows

for a substitution of variables in equation (24),

leading to

epjkgik,j � ÿXa

raGNbasai �t

ap �25�

where

raGN ��raGN�e��2 � �raGN�s��2

q�26�

and

�tap �raGN�e�m

ap � raGN�s�s

ap

raGN: �27�

The right-hand side of equation (25) is the ex-

pression used to de®ne Nye's tensor in equation (9)

relating the gradient of plastic slip to Nye's tensor.

Substitution of equation (25) into equation (21)

using the de®nition of Nye's tensor in equation (9)

yields

kpi � ÿaip � 1

2dpiakk � epjkeelik,j �28�

which is the same expression as equation (16) in theabsence of elastic strain gradients.

The relationships among plastic slip gradients,

geometrically-necessary dislocation density, Nye's

tensor, and lattice curvature presented thus far are

general and apply to any crystal lattice. In the next

section, the presence of GNDs in crystals with a

high degree of symmetry is considered. The sym-

metry allows for multiple dislocation con®gurations

to have the same total geometric properties.Therefore, the crystallographic dislocation state

resultant from the geometric constraints is indeter-

minate. The indeterminacy can be resolved using

two methods, as proposed in the next section.


5. UNIQUE DESCRIPTION OF GNDS INREDUNDANT CRYSTALS

The most rigorous manner to describe crystallo-

graphic dislocation density would be to develop in-

itial conditions and evolution laws for their

densities and their interactions. The geometrical

properties of such dislocation distributions would

just be a consequence of the evolving state. Nye's

tensor would be a simple result of the density at

any instant, and plastic strain would result from the

motion of density through the volume. Such evol-

ution laws on a ®eld basis have not been developed

as of yet, but the plastic slip gradient ®eld imposes

geometric constraints on the dislocation density

state which disallow many crystallographic dislo-

cation distributions.

In crystals with a high degree of symmetry, the

geometric constraints can be satis®ed with many

di�erent dislocation con®gurations due to their

redundancy, much the same as a given plastic de-

formation can be performed by di�erent combi-

nations of slip on individual systems. In such

redundant crystals, the number of distinct dislo-

cation ``types'', each with its own geometric proper-

ties, exceeds the nine independent values in Nye's

tensor, but the concept of geometrically-necessary

dislocations implies a minimization of density.

Consider again Fig. 4 of the dislocation threading

through a reference volume. It was shown that its

total dislocation density could be decomposed into

a geometrically-necessary part and a statistically-

stored part. The geometrically-necessary part was

the minimum dislocation density needed to span the

two endpoints of the dislocation line segment. If the

only information from which crystallographic dislo-

cation densities are to be determined is the result

from geometric constraints, then the crystallo-

graphic dislocation density derived from the con-

straints must be a minimum density. Anything

above the geometric minimum necessary would

(necessarily!) incorporate some statistical character

which cannot be determined through geometric

arguments. By considering di�erent density normal-

izations and minimizing the dislocation density with

respect to those normalizations, a unique descrip-

tion of crystallographic GNDs can be found.

The ®rst step in ®nding the GND con®guration

on a crystallographic basis is to discretize the dislo-

cation space of a crystal. The Burgers vectors in a

crystal are already discrete, but the tangent line vec-

tor of a dislocation is free to occupy any direction

on the slip plane (if dislocations are allowed to

climb, even this restrictive condition no longer

holds!). The tangent line vectors must be discretized

in such a fashion that the discretized space is well

representative of the actual dislocation space. The

discretized Burgers vector, b, and tangent line vec-

tors, t, form n-pairs of geometric dislocation prop-

erties. Nye's tensor, aa, can be written as a

summation of dislocation dyadics, di, premultiplied

by a scalar dislocation density, ri, as follows:

aaa �Xni�1

ridi �29�

where the dislocation dyadic is given by

di � bi ti: �30�Equation (29) can be rewritten such that Nye'stensor, represented as a nine-dimensional columnvector L containing the components of aa, is the

result of a linear operator, A, acting on the n-dimensional crystallographic dislocation densityvector, r, as

Ar � L: �31�The null space of operator A yields those combi-

nations of crystallographic dislocation densitywhich have no geometric consequence; thus thesecombinations can be considered as statistically-

stored. Such dislocation density groups are placedin a subspace of the n-dimensional r-space, rSS;the dimension of the statistically-stored dislocation

density subspace is nÿ 9.In considering GNDs, there are two minimiz-

ations which can be considered. One minimization,

L2, is geometrically motivated through equation (26)and minimizes the sum of the squares of the result-ing dislocation densities. The discretization of the

dislocation space only allows certain dislocations onthe slip plane to exist, and the L2 minimizationtakes this into account by being able to combine

dislocation line lengths into a single dislocation linelength which may not exist within the original dis-cretization. The other minimizing technique, L1, is

energetically motivated. By considering dislocationdensity as line length through a volume, the total

dislocation line energy can be minimized by ®ndingthe dislocation con®guration with the smallest totalline length.

Mathematically, the L2 minimization is the easierof the two methods to compute. The functional,C�r,y�, to be minimized takes the form

C�r,y� � rTr� yT�Arÿ L� �32�where the nine-dimensional vector y contains theLagrange multipliers. The solution can be foundexplicitly because it follows from singular value de-

composition, having the following mathematicalform:

rGN � �ATA�ÿ1ATL � BL �33�which gives an explicit formula for the crystallo-

graphic GND density, rGN, for any given value ofNye's tensor. Furthermore, rGN calculated in thismanner describes a dislocation density subspace

which is orthogonal to rSS such that �rGN�TrSS � 0,and the two subspaces span the total n-dimensionalspace of crystallographic dislocation densities.


There is no explicit formula for determining rGNfrom Nye's tensor with respect to the L1 minimum.

Using this technique, the functional, D�r,y�, to beminimized takes the form

D�r,y� �Xni�1jrij � yT�Arÿ L� �34�

where the nine-dimensional vector y contains the

Lagrange multipliers. A linear simplex method isimplemented to calculate rGN for each di�erentvalue of Nye's tensor. The resultant crystallographicdislocation density vector does not lie outside the

rSS subspace like the one obtained using the L2

minimization. The two techniques are demonstratedon a face-centered cubic (f.c.c.) crystal in the fol-

lowing sections to illustrate the properties of eachnormalization, and the null space of A is also ana-lyzed to determine the properties of rSS in redun-

dant crystals.

6. GEOMETRICALLY-NECESSARY DISLOCATIONSIN F.C.C. CRYSTALS

Face-centered cubic crystals have slip systems in

which the slip planes are of {111} type and the slipdirections are of h110i type. Although any discretebasis of dislocations which exist in the crystal maybe considered, a natural choice is to limit the dislo-

cations to only pure edge and pure screw types asKubin et al. have done in their dislocationsimulations [13]. Adopting this discretization, there

are a total of 18 di�erent dislocation types: 12 edgeand six screw dislocations. The dislocations andtheir line properties are given in Table 1.

Crystallographic dislocation density is described byan 18-dimensional vector in which each distinct dis-location type receives its own index, and ri is the

density of the ith dislocation type. Nye's tensor hasonly nine independent components; therefore, whendescribing the dislocation state in terms of crystallo-

graphic dislocation densities, the problem is under-

de®ned, much like the indeterminacy of apportion-ing slip on crystallographic planes based on aknown plastic deformation.

The geometric properties of the crystallographicdislocation densities were mapped onto the f.c.c.unit cell using equation (9), and the indices of

Nye's tensor became the three orthonormal direc-tions of the f.c.c. unit cell. A set of nine vectors

were found which led to no contribution in Nye'stensor. These null vectors can be considered as theredundant or SSDs and will be discussed in detail

in the next section. The GND density was calcu-lated using the two di�erent techniques outlined inthe previous section, and the resultant GND distri-

butions from the L2 and L1 minimizations will bepresented independently and compared at the end

of this section.Using the singular value decomposition described

above, a set of nine crystallographic dislocation vec-

tors were obtained which minimized the sum of thesquares of the densities and satis®ed the geometricrequirements of Nye's tensor. Table 2 shows the

matrix which can be used to compute a generalcrystallographic dislocation distribution from GND

density using L2 minimization and the SSD densitywith h100i directions as the basis vectors of Nye'stensor. The ®rst nine columns of this matrix make

up the matrix, B, in equation (33). Note that thereare negative crystallographic dislocation densities inthe GND vectors. The negative densities appear as

a result of the discrete dislocation basis chosen. Thebasis de®nes the right-handed edge and screw dislo-

cations as being positive densities, and left-handededge and screw dislocation densities are considerednegative. The L1 formulation employs an expanded

basis, and the necessity to interpret negative den-sities is eliminated.Upon inspection of the nine GND vectors, there

are only two distinct dislocation arrangementswhich are formed: one which exhibits the same

properties as a h100ih100i screw dislocation, andone which exhibits the same properties as a h100i�h010i edge dislocation on the f.c.c. unit cell. The

other seven density vectors are just orthogonaltransformations of these two GND vectors. In f.c.c.materials, dislocations of h100i type are not pre-

ferred, but an arrangement of crystallographic dis-locations can be formed which has the same

geometric properties as a h100i type dislocation.This arrangement for a screw dislocation is math-ematically described by the ®rst density vector (col-

umn 1) in Table 2, and is graphically presented inFig. 6 by interpreting the densities as line lengthswithin a volume described by equation (9). Figure 6

shows the resulting double-helical structure, withscrew dislocations around the perimeter of thestructure and a mix of edge and screw dislocations

in the center. The global geometric properties ofthis dislocation structure are the same as those of a

Table 1. The dislocation basis used to describe the dislocationstate in f.c.c. crystals

Density t s n

r1 1��6p �112� 1��

2p �110� 1��

3p �111�

r2 1��6p �121� 1��

2p �101� 1��

3p �111�

r3 1��6p �211� 1��

2p �011� 1��

3p �111�

r4 1��6p �112 1��

2p �110� 1��

3p �111�

r5 1��6p �121� 1��

2p �101� 1��

3p �111�

r6 1��6p �211� 1��

2p �011� 1��

3p �111�

r7 1��6p �112� 1��

2p �110� 1��

3p �111�

r8 1��6p �121� 1��

2p �101� 1��

3p �111�

r9 1��6p �211� 1��

2p �011� 1��

3p �111�

r10 1��6p �112� 1��

2p �110� 1��

3p �111�

r11 1��6p �121� 1��

2p �101� 1��

3p �111�

r12 1��6p �211� 1��

2p �011� 1��

3p �111�

r13 1��2p �110� 1��

2p �110� 1��

3p �111� or 1��

3p �111�

r14 1��2p �101� 1��

2p �101� 1��

3p �111� or 1��

3p �111�

r15 1��2p �011� 1��

2p �011� 1��

3p �111� or 1��

3p �111�

r16 1��2p �110� 1��

2p �110� 1��

3p �111� or 1��

3p �111�

r17 1��2p �101� 1��

2p �101� 1��

3p �111� or 1��

3p �111�

r18 1��2p �011� 1��

2p �011� 1��

3p �111� or 1��

3p �111�


[100] type screw dislocation. It is periodic in the

direction of the dislocation line vector, [100], and

there is a discrete rotational symmetry about the

[100] axis, much as a �100��100� screw dislocation

would have if it existed in f.c.c. crystals.

A similar arrangement can be created to describe

a h100ih010i edge dislocation in f.c.c. materials.

This arrangement is mathematically described by

the second vector (column 2) in Table 2, and is

graphically shown in Fig. 7. It is made up of 12

edge dislocations, and two screw dislocations which

cross at the center of the structure. The structure is

again periodic in the line direction, and it has the

same global geometric properties as the h100ih010iedge dislocation. The structure also has a mirror

symmetry on either side of the virtual extra atomic

half plane. With these nine vectors, a dislocation

with arbitrary Burgers vector and line direction can

be described as a periodic structure made up of the

18 crystallographic dislocation densities in f.c.c. ma-

terials.

To implement the L1 minimization of GNDs

using the linear simplex method, the discretization

employed for the L2 minimization had to be altered

such that only positive densities were considered.

This was accomplished by giving the left-handed

screw and edge dislocations their own index, i, and

requiring all densities to be positive. The size of the

discrete dislocation basis doubled, introducing 18

more SSD vectors. The new SSD vectors introduced

through this discretization represent dislocation

dipoles formed by right- and left-handed dislo-

cations of equal length. Dislocation dipoles are

what are classically thought to result from plastic

deformation, and are the most common example of

SSDs, and although they were not accounted for in

Table 2. The linear operator used to create geometrically allowed dislocation density from Nye's tensor and statistically storeddislocation density. The ®rst nine columns form the matrix B in equation (33) weighted by the components of Nye's tensor withh100i basis vectors. The last nine columns represent non-dipole components of the statistically-stored dislocation density,

weighted by the parameters li independent of Nye's tensor

Fig. 6. A periodic dislocation network derived through theL2 minimization scheme which has the same geometric

properties as a h100ih100i positive screw dislocation.


the discretized space employed in the L2 formu-

lation, they do appear in this expanded space.

To calculate the GND density using the linear

simplex method, a feasible solution was found

through row reduction of the matrix A from

equation (31) and back substitution. The minimal

L1 density was found through an iterative process,

and had to be conducted for each di�erent value of

Nye's tensor. In order to compare the results of L1

and L2 minimizations, the dislocation structures for

the h100ih100i screw dislocations and the h100ih010iedge dislocations were also found using the L1 mini-

mization technique. The crystallographic dislocation

structure for a �100��100� screw dislocation using the

L1 minimization technique is depicted in Fig. 8. The

structure consists of two separate dislocation lines

with Burgers vectors [110] and �110�, respectively,and the same average tangent line vector [100], and

each threading dislocation line consists of three dis-

location line segments. The crystallographic dislo-

cation structure for the L1 minimal �100��010� edgedislocation found is shown in Fig. 9. It also is made

up of two separate dislocation lines with Burgers

vectors [110] and �110�, respectively, and the same

average tangent line vector [010], and each thread-ing dislocation line consists of two dislocation line

segments.

The purpose of this exercise in crystallographic

GNDs was to ®nd a method to best represent the

actual GND dislocation arrangements in crystals.Both of the methods found periodic dislocation

arrangements which threaded through the reference

volume and had the same global geometric e�ects.

Comparing the screw and edge dislocation arrange-ments obtained with the two di�erent techniques,


properties as a h100ih010i positive edge dislocation.


properties as a h100ih100i positive screw dislocation.


the L1 minimization appears to be a more promis-ing technique than the L2 technique for determining

GNDs. In each case, the L1 method created dislo-cation structures which were composed of two sep-

arate dislocation lines with constant Burgers vectorsthreading through the volume; the arrangements

created by the L2 method were much more compli-cated in the sense that they contained more dislo-

cation segments in their structures and alsorequired the formation of intricate junctions. The

total dislocation line length needed to describe the�100��010� edge dislocation with the L2 minimization

was a factor of 1.25 greater than the correspondingL1 minimization. The total line length needed to

describe the �100��100� screw dislocation with the L2

minimization was a factor of 1.01 greater than the

L1 minimization. Note that this particular crystal-line orientation required the longest total dislo-

cation line length to represent a geometrically-necessary screw dislocation calculated using the L1

minimization. All other crystalline orientations

required shorter total dislocation line lengths. Thedisadvantage of employing the L1 method over theL2 method is that there is no explicit formulation

using the linear simplex method to determine rGN

from Nye's tensor, whereas the singular value de-composition does provide an explicit relation for

the L2 minimum con®guration.

7. STATISTICALLY-STORED DISLOCATIONS INF.C.C. CRYSTALS

Classically, statistically-stored dislocation densityhas been considered to be comprised of dislocation

dipoles and planar dislocation loops. In the formal-ism used to ®nd rGN with the L1 minimization, atotal of 27 SSD density vectors were found: only 18

of these were common dislocation dipoles. In theanalysis of the null space of the linear operator A

in equation (31), nine other SSD structures were

found which are neither simple dislocation dipolesnor simple planar loops. These higher-order closedstructures are mathematically represented by thespace spanned by the last nine vectors of the matrix

in Table 2. An in®nite number of SSD structurescan be created by di�erent combinations of the 27null vectors found, but to illustrate the properties

of higher-order SSD structures, the simplest interms of the fewest number of lines and highestsymmetry have been found. The simplest structure

consists of ®ve dislocation line segments and ismathematically represented, in varying orientations,by the last eight null vectors in Table 2.

Interpreting the densities as dislocation line lengthsas before, the vectors can be graphically rep-


properties as a h100ih010i positive edge dislocation.

Fig. 10. A discrete planar dislocation structure represent-ing the intersection of two dislocation loops with di�erent

Burgers vectors.


resented. The core structure corresponding to the

last eight null vectors is shown in Fig. 10. The

structure consists of a central edge dislocation

which splits into two dislocations which loop

around and connect back to the original center dis-

location. The Burgers vectors of the three dislo-

cation lines which compose this structure are

di�erent, and this structure could be created by the

intersection of two expanding dislocation loops

with di�erent Burgers vectors on the slip plane. The

result would be a dislocation arrangement like the

one depicted. Note, however, that the two dislo-

cations are not required to initially be on the same

slip plane because, through cross-slipping, an inter-

section may also occur. These eight null vectors

span the space of such dislocation interactions on

all four slip planes.

The remaining SSD arrangement in Table 2 is

made up of six edge dislocations in the three-dimen-

sional structure shown in Fig. 11. Three of the dis-

locations are on the same slip plane with the other

three lying, respectively, in the other three slip

planes of the f.c.c. crystal lattice. This structure is

unique in that it contains a three-dimensional stack-ing fault whose boundaries are the edge dislocations

shown in the ®gure. Since the extra half planes ofatoms are fully contained within the structure, thereis no geometric consequence over the volume con-

taining the structure. Linear combinations of thisvector with the other eight can create other, morecomplicated, three-dimensional SSDs. Figure 12

depicts a three-dimensional structure consisting offour screw dislocations on the perimeter of thestructure and four edge dislocations in the interior

of the structure, but as these three-dimensionalstructures become more complicated, it becomesharder to imagine their occurrence in real crystals.In general, the higher-order dislocation structures

in f.c.c. crystals primarily consist of dislocationloop segments which terminate at planar junctions.The total structure self-terminates, which is to say

that the Burgers vector is conserved at each junc-tion in the structure. The closed structures do notthread through the volume as do the crystallo-

graphic GNDs; furthermore, they do not create anydislocations which pierce the surface of the refer-ence volume.

A general dislocation state for f.c.c. crystals inthis basis of 36 linearly independent density vectors(nine crystallographic GND vectors, 18 dislocationdipole vectors, and nine higher-order SSD vectors)

can be constructed by any linear combination ofthe 27 SSD vectors and the particular combinationof GND vectors dictated by the value of Nye's ten-

sor. This geometrically-allowable dislocation(GAD) density would conform to the geometricconstraints, but would not restrict the total crystal-

lographic dislocation density to any minimum prin-ciple. These vectors span the space of all glissiledislocations in f.c.c. crystals and could be used as adiscrete dislocation basis for observed dislocation

arrangements. Also, there may be other crystallinekinetic constraints which may suppress the popu-lation of dislocations on certain planes. The GAD

density vector could be further restricted to re¯ectother such constraints.

8. GEOMETRICALLY-NECESSARY DISLOCATIONSIN POLYCRYSTALLINE AGGREGATES

In this article, parallels have been suggestedbetween the determination of active slip systemsfrom the plastic deformation tensor and the deter-

mination of crystallographic GNDs from Nye's ten-sor. In fact, the method used to determine rGNusing the L1 minimization is analogous to the

method employed by Taylor in the determination ofthe active slip systems in plastic deformation [14, 15].Taylor's method was based on the principle of

maximum work where the active slip systems werethose which were specially oriented and requiredthe least shearing to accomplish the desired defor-mation. The L1 minimization of rGN employs a

Fig. 11. A three-dimensional dislocation structure com-posed of six edge dislocations which self terminates and

has no geometric consequence.


similar energy principle because it produces the dis-

location con®guration with the smallest total line

energy, assuming the constant line energy model,

which satis®es Nye's tensor.

Continuing with this analogy, there is a geometric

relationship between the yield strength of a poly-

crystal and the critical resolved shear strength of a

slip system. There should be a corresponding re-

lationship between Nye's tensor averaged over a

polycrystal and the crystallographic GND density

in a single crystal. The Taylor factor, �m, relating

the tensile yield strength, sy, to the critical resolved

shear strength for slip, tCRSS, acts as an isotropic

interpretation of the crystalline anisotropy at the

continuum level, and is de®ned as

sy � �mtCRSS: �35�The Taylor factor, �m, was successfully derived by

Bishop and Hill, and for f.c.c. materials, �m was

found to be 3.06 [16, 17].

A similar factor may be considered for macro-

scopic strain gradient ®elds in polycrystalline ma-

terials. Of course, GNDs arise in polycrystals in

which macroscopic strain gradients are not present.

Local interactions between neighboring grains give

rise to GNDs leading to the grain-size dependence

of strain hardening [18]. Apart from these local

GNDs, macroscopic strain gradient ®elds lead to a

macroscopic Nye's tensor which must be interpreted

on a continuum level. Using the information at the

single crystal level, a corresponding Nye factor, �r,may be introduced to re¯ect the scalar measure of

geometrically-necessary dislocation density, PGN,

resultant from macroscopic plastic strain gradients.

The simplest macroscopic description of plastic

strain gradient ®elds, which can be described by

single-parameter Nye's tensors, are found in plane

strain bending and torsion. In the calculation of �rfor each of these cases, Nye's tensor was taken to

be uniform across the polycrystal, but the dislo-

cation density was not continuous across the poly-

crystal. These conditions ensured intra-grain lattice

continuity, but allowed dislocation lines to termi-

nate at grain boundaries. The Nye factor was calcu-

lated with the following equation:

�r � 1

nb

Xni�1jrGN�aij;f,y,o �ji �36�

where b is the magnitude of the Burgers vector and

f, y, and o are the Euler angles de®ning the orien-

tation of crystal i, and n is the number of grains in

the sample. The crystallographic GND dislocation

density, rGN, was calculated using the linear simplex

method for each orientation, and the isotropic

measure, �r, of the GND density was the total dislo-

cation line length in each grain. The calculation was

conducted for f.c.c. polycrystals in bending and tor-

sion. In the case of bending, the Nye factor was

found to be �r � 1:85 such that

PGN�e�b � 1:85ja12j �37�and in the case of torsion, the Nye factor was

found to be �r � 1:93 such that

PGN�s�b � 1:93ja11j �38�where the (e) and (s) subscripts denote the isotropic

edge and screw density, respectively.

The Nye factor accounts for the underlying crys-

talline anisotropy in this continuum measure of

GNDs. Current isotropic continuum models [10, 11]

of strain-gradient plasticity use invariants of the

curvature tensor, which is closely related to Nye's

tensor through equation (28), as a measure of

GNDs, but the value of Nye's tensor on a conti-

nuum material point of a polycrystal is not a su�-

cient measure of the GNDs present. The actual

dislocation density based on crystallographic con-

siderations for the case of plane strain bending and

torsion is almost twice that which would be pre-

Fig. 12. A three-dimensional dislocation structure com-posed of four edge and four screw dislocations which self

terminates and has no geometric consequence.


dicted by the use of Nye's tensor alone. Since theplastic resistance of a material is related to the

square root of the dislocation density, the addedresistance due to the presence of GNDs in thesemodels is underestimated.

9. CONCLUDING REMARKS

This article has investigated the geometrical prop-erties of GNDs and SSDs on a crystallographic

dislocation basis. To describe crystallographic geo-metrically-necessary dislocations, a new formulationof Nye's dislocation tensor was introduced, based

on the integrated properties of dislocation lineswithin a reference volume. A discretized version ofthe integral relation in which all crystallographicdislocations could be represented by a ®nite set of

realizable crystallographic dislocation populationswas adopted. The crystallographic GND state wasfound to be underdetermined by the constraints of

Nye's tensor in crystals with a high degree of sym-metry. This redundancy was overcome by employ-ing minimum principles on two di�erent density

normalizations. The analyses were performed on anf.c.c. crystal lattice in which the dislocation popu-lations were limited to either pure edge or pure

screw type, and two categories of dislocationarrangements were found and visualized by consid-ering dislocation density as a line length in avolume. The ®rst category described crystallo-

graphic GNDs through periodic networks whichmaintained the same geometric properties and sym-metries of the material GNDs they represented, and

the second category described SSDs that werehigher-order self-terminating dislocation structureswhich had no geometric consequence. From these

two di�erent types of arrangements, geometrically-allowable dislocation densities could be formulatedwhich conformed to the geometric constraints of

Nye's tensor. The implications of the crystallo-graphic GND arrangements in single crystals on thepresence of GNDs in polycrystalline materials due

to macroscopic plastic strain gradients culminatedin the introduction of a Nye factor to account for

underlying crystalline anisotropy.

AcknowledgementsÐWe would like to thank the NSFunder Grant No. 9700358 and the DoD through theNDSEG Fellowship Program for support of this research.We would also like to thank Ali S. Argon for his manydiscussions on the subject.

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